In general, a typical root of a negative number is complex, so you need to get rid of most roots. A nice approach would be Root, e.g.

Root[ x^3 + 8, #] & /@ Range[3]

{-2, 1 - I Sqrt[3], 1 + I Sqrt[3]}

To get only real roots you can do :

Select[Root[ x^3 + 8, #] & /@ Range[3], Re[#] == # &]

{-2}

This is a handy approach when you have roots of lower orders.

However you'd rather use Reduce or Solve for higher order roots, in this case it works like this :

Reduce[ x^3 + 8 == 0, x]

x == -2 || x == 1 - I Sqrt[3] || x == 1 + I Sqrt[3]

Solve[ x^3 + 8 == 0, x]

{{x -> -2}, {x -> 2 (-1)^(1/3)}, {x -> -2 (-1)^(2/3)}}

To get only real roots one can use for example : Reduce[x^3 + 8 == 0, x, Reals] or Solve[x^3 + 8 == 0, x, Reals]. They do almost the same, but their outputs are a bit different, respectively : in the boolean form and in the form of rules.

As a more appropriate example of what you want to do I could choose this one : (-3)^(1/7). Mathematica treats variables (in general) as complex. So one gets seven roots
and there is the only one real.

Solve[ x^7 + 3 == 0, x, Reals]

{{x -> -3^(1/7)}}

To get the full output one can do this :

points = {Re @ #, Im @ #} & /@ Last @@@ Solve[x^7 + 3 == 0, x]

Absolute values of the roots are the same, so they are found on the circle of a given radius (== 3^(1/7)) :

{ Equal @@ #, radius = #[[1]] } & @ Simplify @ (Norm /@ points)

{True, 3^(1/7)}

To visualize the structure of the output one makes use of ContourPlot of real and imaginary parts of the function (x + I y)^7 + 3 (we write the function explicitly in the complex form since we make plots in real domains of x and y ) :

I quite prefer realPower[x_, r_] := Sign[x]*Abs[x]^r myself. (A similar thing is done in the old package Miscellaneous`RealOnly`.) realPower[-8, 1/3] yields -2, as expected. To get all the real roots, Artes's solution is best.

Mathematica is a registered trademark of Wolfram Research, Inc. While the mark is used herein with the limited permission of Wolfram Research, Stack Exchange and this site disclaim all affiliation therewith.